# Properties

 Label 21.6.a.c Level $21$ Weight $6$ Character orbit 21.a Self dual yes Analytic conductor $3.368$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,6,Mod(1,21)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("21.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 21.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$3.36806021607$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 5 q^{2} + 9 q^{3} - 7 q^{4} + 94 q^{5} + 45 q^{6} - 49 q^{7} - 195 q^{8} + 81 q^{9}+O(q^{10})$$ q + 5 * q^2 + 9 * q^3 - 7 * q^4 + 94 * q^5 + 45 * q^6 - 49 * q^7 - 195 * q^8 + 81 * q^9 $$q + 5 q^{2} + 9 q^{3} - 7 q^{4} + 94 q^{5} + 45 q^{6} - 49 q^{7} - 195 q^{8} + 81 q^{9} + 470 q^{10} + 52 q^{11} - 63 q^{12} - 770 q^{13} - 245 q^{14} + 846 q^{15} - 751 q^{16} - 2022 q^{17} + 405 q^{18} + 1732 q^{19} - 658 q^{20} - 441 q^{21} + 260 q^{22} - 576 q^{23} - 1755 q^{24} + 5711 q^{25} - 3850 q^{26} + 729 q^{27} + 343 q^{28} + 5518 q^{29} + 4230 q^{30} + 6336 q^{31} + 2485 q^{32} + 468 q^{33} - 10110 q^{34} - 4606 q^{35} - 567 q^{36} - 7338 q^{37} + 8660 q^{38} - 6930 q^{39} - 18330 q^{40} - 3262 q^{41} - 2205 q^{42} + 5420 q^{43} - 364 q^{44} + 7614 q^{45} - 2880 q^{46} + 864 q^{47} - 6759 q^{48} + 2401 q^{49} + 28555 q^{50} - 18198 q^{51} + 5390 q^{52} + 4182 q^{53} + 3645 q^{54} + 4888 q^{55} + 9555 q^{56} + 15588 q^{57} + 27590 q^{58} - 11220 q^{59} - 5922 q^{60} - 45602 q^{61} + 31680 q^{62} - 3969 q^{63} + 36457 q^{64} - 72380 q^{65} + 2340 q^{66} + 1396 q^{67} + 14154 q^{68} - 5184 q^{69} - 23030 q^{70} + 18720 q^{71} - 15795 q^{72} + 46362 q^{73} - 36690 q^{74} + 51399 q^{75} - 12124 q^{76} - 2548 q^{77} - 34650 q^{78} + 97424 q^{79} - 70594 q^{80} + 6561 q^{81} - 16310 q^{82} - 81228 q^{83} + 3087 q^{84} - 190068 q^{85} + 27100 q^{86} + 49662 q^{87} - 10140 q^{88} - 3182 q^{89} + 38070 q^{90} + 37730 q^{91} + 4032 q^{92} + 57024 q^{93} + 4320 q^{94} + 162808 q^{95} + 22365 q^{96} + 4914 q^{97} + 12005 q^{98} + 4212 q^{99}+O(q^{100})$$ q + 5 * q^2 + 9 * q^3 - 7 * q^4 + 94 * q^5 + 45 * q^6 - 49 * q^7 - 195 * q^8 + 81 * q^9 + 470 * q^10 + 52 * q^11 - 63 * q^12 - 770 * q^13 - 245 * q^14 + 846 * q^15 - 751 * q^16 - 2022 * q^17 + 405 * q^18 + 1732 * q^19 - 658 * q^20 - 441 * q^21 + 260 * q^22 - 576 * q^23 - 1755 * q^24 + 5711 * q^25 - 3850 * q^26 + 729 * q^27 + 343 * q^28 + 5518 * q^29 + 4230 * q^30 + 6336 * q^31 + 2485 * q^32 + 468 * q^33 - 10110 * q^34 - 4606 * q^35 - 567 * q^36 - 7338 * q^37 + 8660 * q^38 - 6930 * q^39 - 18330 * q^40 - 3262 * q^41 - 2205 * q^42 + 5420 * q^43 - 364 * q^44 + 7614 * q^45 - 2880 * q^46 + 864 * q^47 - 6759 * q^48 + 2401 * q^49 + 28555 * q^50 - 18198 * q^51 + 5390 * q^52 + 4182 * q^53 + 3645 * q^54 + 4888 * q^55 + 9555 * q^56 + 15588 * q^57 + 27590 * q^58 - 11220 * q^59 - 5922 * q^60 - 45602 * q^61 + 31680 * q^62 - 3969 * q^63 + 36457 * q^64 - 72380 * q^65 + 2340 * q^66 + 1396 * q^67 + 14154 * q^68 - 5184 * q^69 - 23030 * q^70 + 18720 * q^71 - 15795 * q^72 + 46362 * q^73 - 36690 * q^74 + 51399 * q^75 - 12124 * q^76 - 2548 * q^77 - 34650 * q^78 + 97424 * q^79 - 70594 * q^80 + 6561 * q^81 - 16310 * q^82 - 81228 * q^83 + 3087 * q^84 - 190068 * q^85 + 27100 * q^86 + 49662 * q^87 - 10140 * q^88 - 3182 * q^89 + 38070 * q^90 + 37730 * q^91 + 4032 * q^92 + 57024 * q^93 + 4320 * q^94 + 162808 * q^95 + 22365 * q^96 + 4914 * q^97 + 12005 * q^98 + 4212 * q^99

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
5.00000 9.00000 −7.00000 94.0000 45.0000 −49.0000 −195.000 81.0000 470.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.6.a.c 1
3.b odd 2 1 63.6.a.b 1
4.b odd 2 1 336.6.a.i 1
5.b even 2 1 525.6.a.b 1
5.c odd 4 2 525.6.d.c 2
7.b odd 2 1 147.6.a.f 1
7.c even 3 2 147.6.e.c 2
7.d odd 6 2 147.6.e.d 2
12.b even 2 1 1008.6.a.a 1
21.c even 2 1 441.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.c 1 1.a even 1 1 trivial
63.6.a.b 1 3.b odd 2 1
147.6.a.f 1 7.b odd 2 1
147.6.e.c 2 7.c even 3 2
147.6.e.d 2 7.d odd 6 2
336.6.a.i 1 4.b odd 2 1
441.6.a.c 1 21.c even 2 1
525.6.a.b 1 5.b even 2 1
525.6.d.c 2 5.c odd 4 2
1008.6.a.a 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 5$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(21))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 5$$
$3$ $$T - 9$$
$5$ $$T - 94$$
$7$ $$T + 49$$
$11$ $$T - 52$$
$13$ $$T + 770$$
$17$ $$T + 2022$$
$19$ $$T - 1732$$
$23$ $$T + 576$$
$29$ $$T - 5518$$
$31$ $$T - 6336$$
$37$ $$T + 7338$$
$41$ $$T + 3262$$
$43$ $$T - 5420$$
$47$ $$T - 864$$
$53$ $$T - 4182$$
$59$ $$T + 11220$$
$61$ $$T + 45602$$
$67$ $$T - 1396$$
$71$ $$T - 18720$$
$73$ $$T - 46362$$
$79$ $$T - 97424$$
$83$ $$T + 81228$$
$89$ $$T + 3182$$
$97$ $$T - 4914$$